The spectrum of the Laplace operator on q-forms of a Kähler manifold
Let M be a compact Kähler manifold of dimension n. Let spec(q,M) be the spectrum of \(\Delta (q)=d\delta +\delta d\), the Laplacian on q-forms. The first author showed previously [Lect. Notes Math. 838, 233-238 (1981; Zbl 0437.53031)] Theorem: Given m, there exists q(m) so if \(spec(q,M)=spec(q,M')\) then M has constant holomorphic sectional curvature h if and only if M' has constant holomorphic sectional curvature - i.e. both have the same local geometry. The proof was based on estimating when certain coefficients in the asymptotic expansion of the heat equation were positive. For example \(q=[m/3]\) if \(m>16\), \(q=0\) if \(m=4,10\), \(q=2\) if \(m=2,6,8,16\), and \(q=3\) if \(m=12\) are suitable choices. In the present paper, the authors list the admissible values of q for \(n\leq 100\). For example if \(n=100\), the suitable intervals are [3,6]\(\cup [26,40]\cup [60,74]\cup [94,97]\) where the latter two intervals arise from the former intervals by duality. (The authors do not, of course, show other values of q fail but only that the asymptotic methods do not suffice).
constant holomorphic sectional curvature
scientific article; zbMATH DE number 4015628
scientific article